$\DeclareMathOperator{\box}{\operatorname{box}}$$\DeclareMathOperator{\cub}{\operatorname{cub}}$ This is a follow up and an extension of another question I asked recently.
A box graph is a graph whose vertices are (closed) boxes of the form $[a_1,b_1]\times \cdots \times [a_n,b_n]$ in $\mathbb R^n$ and there is an edge between two vertices if their corresponding boxes intersect. (For $n=1$, this is also called an interval graph, which was the focus of the above linked question.) The boxicity of a finite graph $G$, denoted as $\box(G)$, is the smallest positive integer $n$ such that $G$ can be realised as a box graph in $\mathbb R^n$. Note that there could be several different box representations of a graph. For a trivial example, given a box representation in $\mathbb R^n$, scaling and translating give other box representations in $\mathbb R^n$. One can use the scaling to ensure that the smallest box is of size $1\times\cdots\times1$.
Let $\mathcal G_\Delta$ be the class of all finite (connected) graphs of maximum degree $\Delta \ge 2$. It is known that all graphs in $\mathcal G_\Delta$ have boxicity at most some $b(\Delta)$ which depends only on $\Delta$. To my knowledge, the best known upper bound is $\Delta \log^{1+o(1)}\Delta$ (see Alex Scott and David R. Wood, Better bounds for poset dimension and boxicity, Trans. Amer. Math. Soc. 373 (2020), 2157-2172).
Furthermore, let $\mathcal G_{n,\Delta}$ be the class of all finite (connected) box graphs in $\mathbb R^n$ with maximum degree $\Delta \ge 2$. (Note that $\mathcal G_{1,\Delta}$ would be the class of all finite (connected) interval graphs of maximum degree $\Delta$, which was denoted as $\mathcal G_\Delta$ in the above linked question. Hope this doesn't cause confusion.)
I want to understand how much we can bound the ratio of the largest and the smallest lengths of the boxes in the box representation of graphs with bounded degree. In particular:
Question 1: For every graph in $\mathcal G_{n,\Delta}$, is there a box representation in $\mathbb R^n$ where all the lengths of all boxes are bounded between $1$ and some function of $n$ and $\Delta$?
For $n=1$, the answer is $2\Delta$, and it is proven in the answer to the above linked question. I originally thought that proving the statement for $n=1$ would be the hard part and extending it to higher dimensions would be easy. But after looking at that answer, I feel like there might be counterexamples already for $n=2$. So here is a weaker question.
Question 2: Are there integers $n(\Delta) \ge 1$ and $L(\Delta) \ge 1$ such that every graph in $\mathcal G_\Delta$ has a box representation in $\mathbb R^{n(\Delta)}$ where all the lengths of all boxes are bounded between $1$ and $L(\Delta)$?
Update: I think I have a (slightly) better way of phrasing Question 2 in more familiar terms.
There is a notion called cubicity of a graph $G$, denoted as $\cub(G)$, which is like boxicity but where the boxes are restricted to be unit cubes. It is known that $\cub(G) \le \box(G) \lceil \log_2 |V(G)| \rceil$ (see Chandran, L. Sunil, Rogers Mathew, and Deepak Rajendraprasad, Upper bound on cubicity in terms of boxicity for graphs of low chromatic number, Discrete Mathematics 339.2 (2016): 443-446). While this does satisfy the requirement of bounded sizes on the boxes, the dimension is not necessarily independent of $|V(G)|$, so it doesn't answer Question 2.
So I want to define an "in between" notion called L-boxicity, denoted as $\box_L(G)$, where I restrict the boxes to have all lengths between $1$ and $L$. It is easy to see that $\cub(G) = \box_1(G)$, $\box(G) = \box_\infty(G)$, and $\box_L(G) \ge \box_{L'}(G)$ whenever $L\le L'$.
I can now rephrase Question 2 as
Question 2': Are there integers $n=n(\Delta)$ and $L=L(\Delta)$ such that for every graph $G\in \mathcal G_\Delta$, $\box_L(G) \le n$?
Any help is much appreciated.
